-5a^2-4a+21=0

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Solution for -5a^2-4a+21=0 equation:


Simplifying
-5a2 + -4a + 21 = 0

Reorder the terms:
21 + -4a + -5a2 = 0

Solving
21 + -4a + -5a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-4.2 + 0.8a + a2 = 0

Move the constant term to the right:

Add '4.2' to each side of the equation.
-4.2 + 0.8a + 4.2 + a2 = 0 + 4.2

Reorder the terms:
-4.2 + 4.2 + 0.8a + a2 = 0 + 4.2

Combine like terms: -4.2 + 4.2 = 0.0
0.0 + 0.8a + a2 = 0 + 4.2
0.8a + a2 = 0 + 4.2

Combine like terms: 0 + 4.2 = 4.2
0.8a + a2 = 4.2

The a term is 0.8a.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8a + 0.16 + a2 = 4.2 + 0.16

Reorder the terms:
0.16 + 0.8a + a2 = 4.2 + 0.16

Combine like terms: 4.2 + 0.16 = 4.36
0.16 + 0.8a + a2 = 4.36

Factor a perfect square on the left side:
(a + 0.4)(a + 0.4) = 4.36

Calculate the square root of the right side: 2.088061302

Break this problem into two subproblems by setting 
(a + 0.4) equal to 2.088061302 and -2.088061302.

Subproblem 1

a + 0.4 = 2.088061302 Simplifying a + 0.4 = 2.088061302 Reorder the terms: 0.4 + a = 2.088061302 Solving 0.4 + a = 2.088061302 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + a = 2.088061302 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + a = 2.088061302 + -0.4 a = 2.088061302 + -0.4 Combine like terms: 2.088061302 + -0.4 = 1.688061302 a = 1.688061302 Simplifying a = 1.688061302

Subproblem 2

a + 0.4 = -2.088061302 Simplifying a + 0.4 = -2.088061302 Reorder the terms: 0.4 + a = -2.088061302 Solving 0.4 + a = -2.088061302 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + a = -2.088061302 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + a = -2.088061302 + -0.4 a = -2.088061302 + -0.4 Combine like terms: -2.088061302 + -0.4 = -2.488061302 a = -2.488061302 Simplifying a = -2.488061302

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.688061302, -2.488061302}

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